**Graduate Studies and Postdoctoral Affairs (GSPA)**

Needles Hall, second floor, room 2201

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For more detailed course information, click on a course title below.

Course ID: 016245

This course studies several classes of methods for the numerical solution of partial differential equations in
multiple dimensions on structured and unstructured grids. Finite volume methods for hyperbolic conservation
laws: linear and nonlinear hyperbolic systems; stability; numerical conservation. Finite element methods for
elliptic and parabolic equations: weak forms; existence of solutions; optimal convergence; higher-order
methods. Examples from fluid and solid mechanics. Additional topics as time permits.

Course ID: 016207

A unified view of linear and nonlinear systems of ordinary differential equations in Rn. Flow operators and
their classification: contractions, expansions, hyperbolic flows. Stable and unstable manifolds. Phase-space
analysis. Nonlinear systems, stability of equilibria and Lyapunov functions. The special case of flows in the
plane, Poincare-Bendixson theorem and limit cycles. Applications to physical problems will be a motivating
influence. (Heldwith AMATH 451)

Course ID: 016208

A thorough discussion of the class of second order linear partial differential equations with constant
coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in
higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and
parabolic equations, continuous dependence results, uniqueness results (including consideration of
unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations
-- Green functions, conformal mapping; hyperbolic equations -- generalized d'Alembert solution, spherical
means, method of descent; transform methods -- Fourier, multiple Fourier, Laplace, Hankel (for all three types
of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations.
(Heldwith AMATH 453)

Course ID: 011278

Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation. (Heldwith AMATH 455)

Course ID: 016209

Concept of functional and its variations. The solution of problems using variational methods - the Euler-
Lagrange equations. Applications include an introduction to Hamilton's principle and optimal control.
(Heldwith AMATH 456)

Course ID: 011279

Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow. (Heldwith AMATH 463)

Course ID: 011280

The Hilbert space of states, observables, and time evolution. Feynman path integral and Greens functions. Approximation methods. Co-ordinate transformations, angular momentum, and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem, and decoherence. Multiparticle quantum mechanics. Bell inequality and basics of quantum computing.

Course ID: 016118

Theory of correlations and entanglement, theory of quantum channels, detectors and the measurement problem in quantum mechanics, phase space formulation of quantum mechanics and entanglement in infinite dimensional quantum systems, introduction to open quantum systems, exploration of current research directions in quantum information.

Course ID: 011281

Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models.

Course ID: 016210

Random variables, expectations, conditional probabilities, conditional expectations, convergence of a
sequence of random variables, limit theorems, minimum mean square error estimation, the orthogonality
principle, random process, discrete-time and continuous-time Markov chains and applications, forward and
backward equation, invariant distribution, Gaussian process and Brownian motion, expectation maximization
algorithm, linear discrete stochastic equations, linear innovation sequences, Kalman filter, various
applications. (Heldwith AMATH 477)

Course ID: 000116

Basic concepts of functional analysis. Topics include: theory of linear operators, nonlinear operators and the Frechet derivative, fixed point theorems, approximate solution of operator equations, Hilbert space, spectral theory. Applications from various areas will be used to motivate and illustrate the theory. A previous undergraduate course in real analysis is strongly recommended.

Course ID: 011283

Elements of asymptotic analysis. Techniques of perturbation theory such as Poincare-Lindstedt, matched asymptotic expansions and multiple scales. Applications to various areas form an essential aspect of the course. Previous courses in real analysis and differential equations at the undergraduate level are strongly recommended.

Course ID: 012670

Introduction to basic algorithms and techniques for numerical computing. Error analysis, interpolation (including splines), numerical differentiation and integration, numerical linear algebra (including methods for linear systems, eigenvalue problems, and the singular value decomposition), root finding for nonlinear equations and systems, numerical ordinary differential equations, and approximation methods (including least squares, orthogonal polynomials, and Fourier transforms).

Course ID: 000724

Discretization methods for partial differential equations, including finite difference, finite volume and finite element methods. Application to elliptic, hyperbolic and parabolic equations. Convergence and stability issues, properties of discrete equations, and treatment of non-linearities. Stiffness matrix assembly and use of sparse matric software. Students should have completed a course in numerical computation at the undergraduate level.

Course ID: 000118

Qualitative theory of systems of ODEs. Topics include: existence/uniqueness of solutions, comparison principle, iterative techniques, stability and boundedness, Lyapunov method, periodic solutions, Floquet theory and Poincare maps, hyperbolicity, stable, unstable and center manifolds, structural stability and bifurcation. Applications from various areas will be used to motivate and illustrate the theory. A previous course in ordinary differential equations at the undergraduate level is strongly recommended.

Course ID: 000119

The main themes are well-posedness of problems, Hilbert space methods, variational principles and integral equation methods. Topics include: first-order nonlinear partial differential equations, quasilinear hyperbolic systems, potential theory, eigenfunctions and eigenvalues, semi-groups, and power series solutions. Applications from various areas will be used to motivate and illustrate the theory. A previous course in partial differential equations at the undergraduate level is strongly recommended.

Course ID: 011284

Basic concepts and classification of stochastic processes. Stochastic differentiation and integration, Markov processes, Chapman-Kolmogorov equation, Fokker-Planck equation, Master equations: mesoscopic vs. macroscopic description. Spectral representation of stationary processes. Correlation function theory. A previous course in probability theory at the undergraduate level is strongly recommended.

Course ID: 016229

Theory and practice of a selection of advanced numerical methods for computational and data sciences.
Algorithms for eigenvalues and singular value decomposition. Multigrid methods for linear and nonlinear
systems. Sparse optimization and compressed sensing. Low-rank tensor and matrix decomposition. Nonlinear
convergence acceleration. Randomized numerical linear algebra. Adjoint methods and automatic
differentiation for neural networks and optimal control. Stochastic gradient descent and variants. Efficient
computer implementation of the algorithms and applications with real-world data. Students should have
completed an introductory course on numerical methods.

Course ID: 016230

Mathematical theory of finite element methods, with error analysis and stability. Continuous and
discontinuous Galerkin methods and mixed finite element methods, applied to elliptic, parabolic, and
hyperbolic partial differential equations including compressible and/or incompressible Euler and Navier-Stokes
equations. Implementation in two and three dimensions. Students should have completed an introductory
course on numerical methods for partial differential equations.

Course ID: 011285

Concepts of stability and boundedness, basic stability criteria, comparison methods, large scale systems, method of decomposition and aggregation, method of several Lyapunov functions, method of vector Lyapunov functions, method of higher derivatives. Stability problems in ecology, mechanics, neural networks and control systems. Students should have completed AM751 or equivalent.

Course ID: 000132

The main theme is the extension of control theory beyond systems modelled by linear ordinary differential equations. Topics include: advanced systems theory, control of nonlinear systems, control of partial differential equations and delay equations. Students should have completed an introductory undergraduate course in control theory.

Course ID: 000133

Mathematical methods, stability of parellel flows for unstratified and stratified fluids, Rayleigh-Taylor instability, centrifugal instability, barotropic and baroclinic instabilities, the effects of viscosity and the Orr-Sommerfeld equation, transition to turbulence, averaged equations, closure problem, homogeneous isotropic turbulence, turbulent boundary layers, effects of stratification. Students should have completed an introductory undergraduate course in fluid mechanics.

Course ID: 000134

Dispersive waves, propagation of dispersive waves in an inhomogeneous medium (WKB theory). Nonlinear resonant interactions. Solitons: completely integrable nonlinear wave equations (e.g., the KdV equation, nonlinear Schrodinger equations) and the inverse Scattering Transform. Applications to water waves and nonlinear optics. Introducation to weakly nonlocal solitary waves and beyond-all-orders asymptotics. Completion of an upper year course in partial differential equations is strongly recommended.

Course ID: 011589

Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.

Course ID: 012151

Introduction to scalar field theory and its canonical quantization in flat and curved spacetimes. The flat space effects of Casimir and Unruh. Quantum fluctuations of scalar fields and of the metric on curved space-times and application to inflationary cosmology. Hawking radiation.

Course ID: 000135

Review of relativistic quantum mechanics and classical field theory. Quantization of free quantum fields (the particle interpretation of field quanta). Canonical quantization of interacting fields (Feynman rules). Application of the formalism of interacting quantum fields to lowestÂ¿order quantum electrodynamic processes. Radiative corrections and renormalization.

Course ID: 010439

Review of elementary general relativity. Timelike and null geodesic congruences. Hypersurfaces and junction conditions. Lagrangian and Hamiltonian formulations of general relativity. Mass and angular momentum of a gravitating body. The laws of black-hole mechanics.

Course ID: 011282

Introduction to the differential geometry of Lorentzian manifolds. The priniciples of general relativity. Causal structure and cosmological singularities. Cosmological space-times with Killing vector fields. Friedmann-Lemaitre cosmologies, scalar, vector and tensor perburbations in the linear and nonlinear regimes. De Sitter space-times and inflationary models.

Course ID: 012567

Review of the axioms of quantum theory and derivation of generalized axioms by considering states, transformations, and measurements in an extended Hilbert space. Master equations and the Markov approximation. Standard models of system-environment interactions and the phenomenology of decoherence. Introduction to quantum control with applications in NMR, quantum optics, and quantum computing.

Course ID: 000136

Review mathematical formulation of operational quantum theory; theory of measurements and decoherence; quantum-classical contrast; review of historical perspectives on interpretation, including EPR paradox; Bell's theorem, non-locality and contextuality; PBR theorem; selected topics including overviews of current interpretations of quantum mechanics and critical experiments in quantum foundations.

Course ID: 013468

Biological and clinical aspects of cancer. Overview of recent mathematical models developed to examine different stages of cancer growth and therapeutic strategies, including ordinary and partial differential equation models, discrete models and models based on continuum mechanics. Various analytical and numerical methods will be used in the analysis of these models.

Course ID: 013469

Dynamic mathematical modelling of biological process at the cellular level. Intracellular networks: metabolism, signal transduction, and genetic regulatory networks. Neural networks: from biophysical modelling of single neurons to the analysis of network behaviour. Modelling will be carried out primarily through ordinary differential equations; analysis will involve application of dynamical systems tools and simulations. Other relevant modelling frameworks (PDEs, delay equations, stochastic methods) will be touched on as time allows.

Needles Hall, second floor, room 2201

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land promised to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.